1Which of these has the same dimensions as angle?
Angle and coefficient of friction both have the dimensions [M⁰ L⁰ T⁰].
MechanicsClass 11, System of Particles and Rotational Motion
Dimensional formula of Angle
[M0 L0 T0]
Arc length divided by radius, so the lengths cancel. An angle is dimensionless, which is why sin and cos can take it.
How to get the dimensional formula of angle
- Start from the definition: angle = length / length.
- Put in the dimension of each part: [L] / [L].
- Add the powers of each base quantity: [M0 L0 T0].
Quantities with the dimensions [M⁰ L⁰ T⁰]
- Solid angle
- Coefficient of friction
- Coefficient of restitution
- Relative density
- Strain
- Poisson's ratio
- Reynolds number
- Emissivity
- Ratio of specific heats
- Sound level
- Relative permittivity
- Magnetic susceptibility
- Relative permeability
- Power factor
- Quality factor
- Refractive index
- Magnification
Equal dimensions do not make them the same quantity. Compare angle with the others in the full grid.
Three questions on angle
2Which of these has no dimensions at all?
Angle is a ratio, so every power in its dimensional formula is zero.
3Which one has different dimensions from the others?
Energy density is [M L⁻¹ T⁻²]. The other three are all [M⁰ L⁰ T⁰].
An equation that uses angle
Equation checker
Matching dimensions are necessary, not sufficient. A check cannot see pure numbers such as 1/2 or 2π, so a consistent equation can still be wrong.
Read as
W = F s cos(θ)W[M L2 T-2]=F s cos(theta)[M L2 T-2]Dimensionally consistentEvery term is [M L2 T-2]
θ = angle
Angle: quick answers
What is the dimensional formula of angle?
The dimensional formula of angle is [M⁰ L⁰ T⁰]. It comes from angle = length / length.
What is the SI unit of angle?
The SI unit of angle is the radian (rad).
Is angle dimensionless?
Yes. Angle is a ratio of like quantities, so every power in its dimensional formula is zero: [M⁰ L⁰ T⁰].